Experiments and modeling of nonlinear frequency response of oscillations of a sessile droplet subjected to horizontal vibrations

Abstract.

In this paper, we experimentally studied the response frequency of oscillations of a sessile water droplet, subjected to horizontal vibrations at varying excitation frequency (5-250 Hz and 40 kHz) and amplitude (0.015 mm to 0.5 mm for low frequencies and 600nm for ultrasonic frequency), as well as static contact angle of the glass substrate ($30^{\circ}$, $75^{\circ}$ , $90^{\circ}$, $115^{\circ}$). The droplets were pinned during the experiments and non-axisymmetric oscillation modes were excited due to the horizontal vibrations. For the first time, we observed that at a sufficiently high vibration amplitude, when the excitation frequency is lower than the smallest natural frequency of the sessile droplet, the droplet oscillates at a response frequency multiple of the excitation frequency. At higher excitation frequencies up to several hundreds of Hz, the droplet oscillates nearly at the excitation frequency. At ultrasonic excitation frequency, however, the droplet cannot follow the excitations, since there is a physical limitation for forming infinite modes (infinite wavenumber) on the surface of a small droplet. We have modeled these behaviors with a nonlinear mass-spring-damper system by combining two established models: the Duffing and Van der Pol equations, in order to simulate both nonlinear damping and stiffness.

Graphical abstract